Existence of solution for a system involving fractional Laplacians and a Radon measure

Abstract

An existence of a nontrivial solution in some `weaker' sense of the following system of equations \begin{align*} (-Δ)^{s}u+l(x)ϕu+w(x)|u|^{k-1}u&=μ~\text{in}~Ω\nonumber\\ (-Δ)^{s}ϕ&= l(x)u^2~\text{in}~Ω\nonumber\\ u=ϕ&=0 ~\text{in}~\mathbb{R}^N\setminusΩ\end{align*} has been proved. Here $s\in (0,1)$, $l,w$ are bounded nonnegative functions in $\Omega$, $\mu$ is a Radon measure and $k>1$ belongs to a certain range.